2,891 research outputs found
Maximizing five-cycles in Kr-free graphs
The Erdos Pentagon problem asks to find an n-vertex triangle-free graph that is maximizing the number of 5-cycles. The problem was solved using flag algebras by Grzesik and independently by Hatami, Hladky, Kral, Norin, and Razborov. Recently, Palmer suggested the general problem of maximizing the number of 5-cycles in K_{k+1}-free graphs. Using flag algebras, we show that every K_{k+1}-free graph of order n contains at most 110k4(k4−5k3+10k2−10k+4)n5+o(n5)
copies of C_5 for any k≥3, with the Turan graph begin the extremal graph for large enough n
Lower Bounds for Symmetric Circuits for the Determinant
Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent. The symmetry restriction is that the circuits which take a matrix input are unchanged by a permutation applied simultaneously to the rows and columns of the matrix. Under such restrictions we have polynomial-size circuits for computing the determinant but no subexponential size circuits for the permanent. Here, we consider a more stringent symmetry requirement, namely that the circuits are unchanged by arbitrary even permutations applied separately to rows and columns, and prove an exponential lower bound even for circuits computing the determinant. The result requires substantial new machinery. We develop a general framework for proving lower bounds for symmetric circuits with restricted symmetries, based on a new support theorem and new two-player restricted bijection games. These are applied to the determinant problem with a novel construction of matrices that are bi-adjacency matrices of graphs based on the CFI construction. Our general framework opens the way to exploring a variety of symmetry restrictions and studying trade-offs between symmetry and other resources used by arithmetic circuits
Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap
Let \phi(G) be the minimum conductance of an undirected graph G, and let
0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the
normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2,
\phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee
is achieved by the spectral partitioning algorithm. This improves Cheeger's
inequality, and the bound is optimal up to a constant factor for any k. Our
result shows that the spectral partitioning algorithm is a constant factor
approximation algorithm for finding a sparse cut if \lambda_k$ is a constant
for some constant k. This provides some theoretical justification to its
empirical performance in image segmentation and clustering problems. We extend
the analysis to other graph partitioning problems, including multi-way
partition, balanced separator, and maximum cut
Symmetric Arithmetic Circuits.
We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry
restriction. In the context of circuits computing polynomials defined on a matrix of variables, such
as the determinant or the permanent, the restriction amounts to requiring that the shape of the
circuit is invariant under row and column permutations of the matrix. We establish unconditional,
nearly exponential, lower bounds on the size of any symmetric circuit for computing the permanent
over any field of characteristic other than 2. In contrast, we show that there are polynomial-size
symmetric circuits for computing the determinant over fields of characteristic zero
Choiceless Polynomial Time, Symmetric Circuits and Cai-F\"urer-Immerman Graphs
Choiceless Polynomial Time (CPT) is currently the only candidate logic for
capturing PTIME (that is, it is contained in PTIME and has not been separated
from it). A prominent example of a decision problem in PTIME that is not known
to be CPT-definable is the isomorphism problem on unordered
Cai-F\"urer-Immerman graphs (the CFI-query). We study the expressive power of
CPT with respect to this problem and develop a partial characterisation of
solvable instances in terms of properties of symmetric XOR-circuits over the
CFI-graphs: The CFI-query is CPT-definable on a given class of graphs only if:
For each graph , there exists an XOR-circuit , whose input gates are
labelled with edges of , such that is sufficiently symmetric with
respect to the automorphisms of and satisfies certain other circuit
properties. We also give a sufficient condition for CFI being solvable in CPT
and develop a new CPT-algorithm for the CFI-query. It takes as input structures
which contain, along with the CFI-graph, an XOR-circuit with suitable
properties. The strongest known CPT-algorithm for this problem can solve
instances equipped with a preorder with colour classes of logarithmic size. Our
result implicitly extends this to preorders with colour classes of
polylogarithmic size (plus some unordered additional structure). Finally, our
work provides new insights regarding a much more general problem: The existence
of a solution to an unordered linear equation system over a
finite field is CPT-definable if the matrix has at most logarithmic rank
(with respect to the size of the structure that encodes the equation system).
This is another example that separates CPT from fixed-point logic with
counting
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